[Turkmath:3340] MSGSÜ-Matematik Genel Seminer- Michel Lavrauw- 01.11.2018, 16:00

[Sibel Şahin]

Sayın liste üyeleri,

*1 Kasım Perşembe 16:00'**da* MSGSÜ Matematik Bölümü Genel Semineri'nde
Sabancı Üniversitesi Mühendislik ve Doğa Bilimleri Fakültesi'nden  *Michel
Lavrauw*  " *Pencils of conics over finite fields**" *başlıklı bir konuşma
verecektir. Konuşmanın özeti aşağıda yer almaktadır.

Seminerde görüşmek dileğiyle,
Sibel ŞAHİN

*Başlık:* *Pencils of conics over finite fields*

*Özet : *A pencil of quadrics is a two-dimensional subspace of the vector
space of quadrics of a given projective space, and is therefore defined by
a pair of quadratic forms on a given vector space. The study of pencils of
quadrics dates back to the late 19th century when Weierstrass (1868)
studied the irrational form of a regular pencil and Kronecker (1890)
discussed the singular case (both over the complex numbers). The methods
developed by Weierstrass and Kronecker resulted in the description of
pencils of quadrics using the theory of invariant factors and elementary
divisors. Dickson was the first to study pencils over finite fields (for
which the aforementioned method fails). In 1908 Dickson classified pencils
of quadrics in projective planes (i.e. conics) over finite fields of odd
characteristic, giving explicit coordinate transformations in order to
reduce the families of ternary quadratic forms to canonical representatives
of the associated equivalence classes of pencils. Part of his proof relies
on the knowledge of the number of irreducible cubics of a given form and
refers to Dickson's treatise on Linear Groups. Recently, together with
Tomasz Popiel, we completed the classification of pencils of conics over
finite fields of even characteristic.

In this talk I will explain how this classification follows from our work
on the symmetric representation of orbits of lines (under the natural
action of GL(3,K)xGL(3,K)) in the 2-fold tensor product of 3-dimensional
vector spaces. Our proof is mostly geometric and works for finite fields of
both even and odd characteristic, and so in particular we recover Dickson's
classification. The proof also works for the reals and all algebraically
closed fields (the approach using elementary divisors also fails for
algebraically closed fields of characteristic 2). Particularly interesting
is the case of pencils without degenerate conics (a case which does not
appear if the field is algebraically closed). Our results imply that each
two such pencils are projectively equivalent.
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